We implement the Bogoliubov-de Gennes (BdG) equation in real-space using the screened Korringa-Kohn-Rostoker (KKR) method. This allows us to solve, self-consistently, the superconducting state for 3d crystals including substitutional impurities with a full normal-state DFT band structure. We apply the theoretical framework to bulk Nb with impurities. Without impurities, Nb has an anisotropic gap structure with two distinct peaks around the Fermi level. In the presence of non-magnetic impurities those peaks are broadened due to the scattering between the two bulk superconducting gaps, however the peaks remain separated. As a second example of self-consistent real-space solutions of the BdG equations we examine superconducting clusters embedded within a non-superconducting bulk metallic host. This allows us to estimate the coherence length of the superconductor and we show that, within our framework, the coherence length of the superconductor is related to the inverse of the gap size, just as in bulk BCS theory.
Bibliographical note9pages, 15 figures